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March  2020, 28(1): 27-46. doi: 10.3934/era.2020003

## Blow-up criterion for the 3D viscous polytropic fluids with degenerate viscosities

 1 School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai 200240, China 2 School of Mathematics, Georgia Institute of Technology, Atlanta 30332, USA

* Corresponding author

Received  September 2019 Revised  September 2019 Published  March 2020

Fund Project: The first author is supported in part by China Scholarship Council 201806230126 and National Natural Science Foundation of China under grant 11571232.

In this paper, the Cauchy problem of the $3$D compressible Navier-Stokes equations with degenerate viscosities and far field vacuum is considered. We prove that the $L^\infty$ norm of the deformation tensor $D(u)$ ($u$: the velocity of fluids) and the $L^6$ norm of $\nabla \log \rho$ ($\rho$: the mass density) control the possible blow-up of regular solutions. This conclusion means that if a solution with far field vacuum to the Cauchy problem of the compressible Navier-Stokes equations with degenerate viscosities is initially regular and loses its regularity at some later time, then the formation of singularity must be caused by losing the bound of $D(u)$ or $\nabla \log \rho$ as the critical time approaches; equivalently, if both $D(u)$ and $\nabla \log \rho$ remain bounded, a regular solution persists.

Citation: Yue Cao. Blow-up criterion for the 3D viscous polytropic fluids with degenerate viscosities. Electronic Research Archive, 2020, 28 (1) : 27-46. doi: 10.3934/era.2020003
##### References:
 [1] J. Beale, T. Kato and A. Majda, Remarks on the breakdown of smooth solutions for the $3$-D Euler equations, Commun. Math. Phys., 94 (1984), 61-66.  doi: 10.1007/BF01212349.  Google Scholar [2] Y. Cho, H. Choe and H. Kim, Unique solvability of the initial boundary value problems for compressible viscous fluids, J. Math. Pure. Appl., 83 (2004), 243-275.  doi: 10.1016/j.matpur.2003.11.004.  Google Scholar [3] M. Ding and S. Zhu, Vanishing viscosity limit of the Navier-Stokes equations to the Euler equations for compressible fluid flow with far field vacuum, J. Math. Pure. Appl., 107 (2017), 288-314.  doi: 10.1016/j.matpur.2016.07.001.  Google Scholar [4] E. Feireisl, A. Novotný and H. Petzeltová, On the existence of globally defined weak solutions to the Navier-Stokes equations, J. Math. Fluid Mech., 3 (2001), 358-392.  doi: 10.1007/PL00000976.  Google Scholar [5] G. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations, Steady-state problems. 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Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, American Mathematical Society, Providence, RI 1968.  Google Scholar [10] Y. Li, R. Pan and S. Zhu, Recent progress on classical solutions for compressible isentropic Navier-Stokes equations with degenerate viscosities and vacuum, Bulletin of the Brazilian Mathematical Society, 47 (2016), 507-519.  doi: 10.1007/s00574-016-0165-7.  Google Scholar [11] Y. Li, R. Pan and S. Zhu, On classical solutions to 2D shallow water equations with degenerate viscosities, J. Math. Fluid Mech., 19 (2017), 151-190.  doi: 10.1007/s00021-016-0276-3.  Google Scholar [12] Y. Li, R. Pan and S. Zhu, On classical solutions for viscous polytropic fluids with degenerate viscosities and vacuum, Arch. Rational. Mech. Anal., 234 (2019), 1281-1334.  doi: 10.1007/s00205-019-01412-6.  Google Scholar [13] Y. Li and S. Zhu, Existence results and blow-up criterion of compressible radiation hydrodynamic equations, J. Dyn. Differ. 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Repovš, Nonlinear Analysis-Theory and Methods, Springer Monographs in Mathematics. Springer, Berlin, 2019. doi: 10.1007/978-3-030-03430-6.  Google Scholar [19] G. Ponce, Remarks on a paper: Remarks on the breakdown of smooth solutions for the $3$-D Euler equations, Commun. Math. Phys., 98 (1985), 349-353.   Google Scholar [20] O. Rozanova, Blow-up of smooth highly decreasing at infinity solutions to the compressible Navier-Stokes equation, J. Differential Equations, 245 (2008), 1762-1774.  doi: 10.1016/j.jde.2008.07.007.  Google Scholar [21] E. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, Princeton NJ, 1970.   Google Scholar [22] Y. Sun, C. Wang and Z. Zhang, A Beale-Kato-Majda blow-up criterion for the 3-D compressible Navier-Stokes equations, J. Math. Pure. Appl., 95 (2011), 36-47.  doi: 10.1016/j.matpur.2010.08.001.  Google Scholar [23] Z. Xin, Blowup of smooth solutions to the compressible Navier-Stokes equation with compact density, Commun. Pure Appl. Math., 51 (1998), 229-240.  doi: 10.1002/(SICI)1097-0312(199803)51:3<229::AID-CPA1>3.0.CO;2-C.  Google Scholar [24] R. Xu, M. Zhang, S. Chen, Y. Yang and J. Shen, The initial-boundary value problems for a class of sixth order nonlinear wave equation, Discrete and Continuous Dynamical Systems, 37 (2017), 5631-5649.  doi: 10.3934/dcds.2017244.  Google Scholar [25] S. Zhu, Existence results for viscous polytropic fluids with degenerate viscosity coefficients and vacuum, J. Differential Equations, 259 (2015), 84-119.  doi: 10.1016/j.jde.2015.01.048.  Google Scholar [26] S. Zhu, On classical solutions of the compressible magnetohydrodynamic equations with vacuum, SIAM J. Math. Anal., 47 (2015), 2722-2753.  doi: 10.1137/14095265X.  Google Scholar

show all references

##### References:
 [1] J. Beale, T. Kato and A. Majda, Remarks on the breakdown of smooth solutions for the $3$-D Euler equations, Commun. Math. Phys., 94 (1984), 61-66.  doi: 10.1007/BF01212349.  Google Scholar [2] Y. Cho, H. Choe and H. Kim, Unique solvability of the initial boundary value problems for compressible viscous fluids, J. Math. Pure. Appl., 83 (2004), 243-275.  doi: 10.1016/j.matpur.2003.11.004.  Google Scholar [3] M. Ding and S. Zhu, Vanishing viscosity limit of the Navier-Stokes equations to the Euler equations for compressible fluid flow with far field vacuum, J. Math. Pure. Appl., 107 (2017), 288-314.  doi: 10.1016/j.matpur.2016.07.001.  Google Scholar [4] E. Feireisl, A. Novotný and H. Petzeltová, On the existence of globally defined weak solutions to the Navier-Stokes equations, J. Math. Fluid Mech., 3 (2001), 358-392.  doi: 10.1007/PL00000976.  Google Scholar [5] G. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations, Steady-state problems. Second edition. Springer Monographs in Mathematics. Springer, New York, 2011. doi: 10.1007/978-0-387-09620-9.  Google Scholar [6] Y. Geng, Y. Li and S. Zhu, Vanishing viscosity limit of the Navier-Stokes equations to the Euler equations for compressible fluid flow with vacuum, Arch. Rational. Mech. Anal., 234 (2019), 727-775.  doi: 10.1007/s00205-019-01401-9.  Google Scholar [7] X. Huang, J. Li and Z. Xin, Blowup criterion for viscous barotropic flows with vacuum states, Commum. Math. Phys., 301 (2011), 23-35.  doi: 10.1007/s00220-010-1148-y.  Google Scholar [8] X. Huang, J. Li and Z. Xin, Global well-posedness of classical solutions with large oscillations and vacuum to the three-dimensional isentropic compressible Navier-Stokes equations, Commun. Pure. Appl. Math., 65 (2012), 549-585.  doi: 10.1002/cpa.21382.  Google Scholar [9] O. A. Ladyzenskaja and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, American Mathematical Society, Providence, RI 1968.  Google Scholar [10] Y. Li, R. Pan and S. Zhu, Recent progress on classical solutions for compressible isentropic Navier-Stokes equations with degenerate viscosities and vacuum, Bulletin of the Brazilian Mathematical Society, 47 (2016), 507-519.  doi: 10.1007/s00574-016-0165-7.  Google Scholar [11] Y. Li, R. Pan and S. Zhu, On classical solutions to 2D shallow water equations with degenerate viscosities, J. Math. Fluid Mech., 19 (2017), 151-190.  doi: 10.1007/s00021-016-0276-3.  Google Scholar [12] Y. Li, R. Pan and S. Zhu, On classical solutions for viscous polytropic fluids with degenerate viscosities and vacuum, Arch. Rational. Mech. Anal., 234 (2019), 1281-1334.  doi: 10.1007/s00205-019-01412-6.  Google Scholar [13] Y. Li and S. Zhu, Existence results and blow-up criterion of compressible radiation hydrodynamic equations, J. Dyn. Differ. Equ., 29 (2017), 549-595.  doi: 10.1007/s10884-015-9455-9.  Google Scholar [14] W. Lian and R. Xu, Global well-posedness of nonlinear wave equation with weak and strong damping terms and logarithmic source term, Adv. Nonlinear Anal., 9 (2020), 613-632.  doi: 10.1515/anona-2020-0016.  Google Scholar [15] W. Lian, V. D. Rǎdulescu, R. Xu, Y. Yang and N. Zhao, Global well-posedness for a class of fourth-order nonlinear strongly damped wave equations, Advances in Calculus of Variations, 2019. doi: 10.1515/acv-2019-0039.  Google Scholar [16] P. Lions, Mathematical Topics in Fluid Mechanics: Compressible Models, Oxford University Press, USA, 1996.   Google Scholar [17] A. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Appl. Math. Sci., vol. 53, Springer, New York, 1984. doi: 10.1007/978-1-4612-1116-7.  Google Scholar [18] N. S. Papageorgiou, V. D. Rǎdulescu and D. D. Repovš, Nonlinear Analysis-Theory and Methods, Springer Monographs in Mathematics. Springer, Berlin, 2019. doi: 10.1007/978-3-030-03430-6.  Google Scholar [19] G. Ponce, Remarks on a paper: Remarks on the breakdown of smooth solutions for the $3$-D Euler equations, Commun. Math. Phys., 98 (1985), 349-353.   Google Scholar [20] O. Rozanova, Blow-up of smooth highly decreasing at infinity solutions to the compressible Navier-Stokes equation, J. Differential Equations, 245 (2008), 1762-1774.  doi: 10.1016/j.jde.2008.07.007.  Google Scholar [21] E. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, Princeton NJ, 1970.   Google Scholar [22] Y. Sun, C. Wang and Z. Zhang, A Beale-Kato-Majda blow-up criterion for the 3-D compressible Navier-Stokes equations, J. Math. Pure. Appl., 95 (2011), 36-47.  doi: 10.1016/j.matpur.2010.08.001.  Google Scholar [23] Z. Xin, Blowup of smooth solutions to the compressible Navier-Stokes equation with compact density, Commun. Pure Appl. Math., 51 (1998), 229-240.  doi: 10.1002/(SICI)1097-0312(199803)51:3<229::AID-CPA1>3.0.CO;2-C.  Google Scholar [24] R. Xu, M. Zhang, S. Chen, Y. Yang and J. Shen, The initial-boundary value problems for a class of sixth order nonlinear wave equation, Discrete and Continuous Dynamical Systems, 37 (2017), 5631-5649.  doi: 10.3934/dcds.2017244.  Google Scholar [25] S. Zhu, Existence results for viscous polytropic fluids with degenerate viscosity coefficients and vacuum, J. Differential Equations, 259 (2015), 84-119.  doi: 10.1016/j.jde.2015.01.048.  Google Scholar [26] S. Zhu, On classical solutions of the compressible magnetohydrodynamic equations with vacuum, SIAM J. Math. Anal., 47 (2015), 2722-2753.  doi: 10.1137/14095265X.  Google Scholar
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